We present an efficient semiparametric variational method to approximate the Gibbs posterior distribution of Bayesian regression models, which predict the data through a linear combination of the available covariates. Remarkable cases are generalized linear mixed models, support vector machines, quantile and expectile regression. The variational optimization algorithm we propose only involves the calculation of univariate numerical integrals, when no analytic solutions are available. Neither differentiability, nor conjugacy, nor elaborate data-augmentation strategies are required. Several generalizations of the proposed approach are discussed in order to account for additive models, shrinkage priors, dynamic and spatial models, providing a unifying framework for statistical learning that cover a wide range of applications. The properties of our semiparametric variational approximation are then assessed through a theoretical analysis and an extensive simulation study, in which we compare our proposal with Markov chain Monte Carlo, conjugate mean field variational Bayes and Laplace approximation in terms of signal reconstruction, posterior approximation accuracy and execution time. A real data example is then presented through a probabilistic load forecasting application on the US power load consumption data.

翻译：本文提出一种高效半参数变分方法，用于近似贝叶斯回归模型的吉布斯后验分布。该类模型通过可用协变量的线性组合对数据进行预测，重要特例包括广义线性混合模型、支持向量机、分位数回归及期望分位数回归。所提出的变分优化算法在无解析解时仅需计算一元数值积分，且无需满足可微性、共轭性或复杂数据增广策略条件。为涵盖加性模型、收缩先验、动态模型及空间模型，本文探讨了所提方法的多种推广形式，构建了适用于广泛应用的统计学习统一框架。通过理论分析与大规模仿真研究评估了该半参数变分近似的性能，在信号重建、后验近似精度及计算时间三个维度上，将本方法与马尔可夫链蒙特卡洛、共轭平均场变分贝叶斯及拉普拉斯逼近进行对比。最后通过美国电力负荷消耗数据的概率负荷预测实例进行实证分析。